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If \(n = x^5*y^7\), where \(x\) and \(y\) are positive integers greater than 1, then how many positive divisors does \(n\) have?
(1) \(x\) does not have a factor \(p\) such that \(1 < p < x\) and \(y\) does not have a factor \(q\) such that \(1 < q < y\). This statement implies that both \(x\) and \(y\) are primes (a prime number does not have a factor which is greater than 1 and less than itself, it has only two factors 1 and itself). Now, if \(x\) and \(y\) are different primes, then the number of factors of \(n\) will be \((5+1)(7+1)=48\) but if \(x=y\), then \(n = x^5*y^7=x^{12}\) and it will have 13 factors. Not sufficient.
(2) \(n\) has only two prime factors. If those primes are \(x\) and \(y\), then the number of factors of \(n\) will be \((5+1)(7+1)=48\) but if, say \(x=2*3=6\) and \(y=2\), then \(n = x^5*y^7=2^{12}*3^5\) and it will have \((12+1)(5+1)=78\) factors. Not sufficient.
(1)+(2) Since from (2) \(n\) has only two prime factors, then from (1) it follows that \(x\) and \(y\) are different primes so the number of factors of \(n\) will be \((5+1)(7+1)=48\). Sufficient.
What about when x is exactly 1? That satisfies statement one (x has no factor greater than 1 and less than itself), but would lead to answer E eventually.
What about when x is exactly 1? That satisfies statement one (x has no factor greater than 1 and less than itself), but would lead to answer E eventually.
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Thank you. Edited to rule out this case.
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Statement 1 raises the fact that x and y could be different or the same. statement 2 says that n has two prime factors. when the statements are combined together it is possible for x is equal to 6 and y is 2. Here the n has two prime factors and they are different. Also x could equal 2 and y equal 3. Both scenarios give us different factors. The answer should be E.
Statement 1 raises the fact that x and y could be different or the same. statement 2 says that n has two prime factors. when the statements are combined together it is possible for x is equal to 6 and y is 2. Here the n has two prime factors and they are different. Also x could equal 2 and y equal 3. Both scenarios give us different factors. The answer should be E.
The answer should be and is C, not E.
(1) means that x and y are primes, so x cannot be 6. 6 does have factors which are more than 1 and less than 6 itself: 2 and 3.
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Is there a place where these questions are just listed? I notice that often the title of the post is something like "Mxy-zw" - are they being taken from a book or something? And how do I get my hands on it?
My initial instinct in seeing two variables x and y is that they are different values. Based on this answer explanation I should completely scrub this line of thinking from my mind for the GMAT, right?
My initial instinct in seeing two variables x and y is that they are different values. Based on this answer explanation I should completely scrub this line of thinking from my mind for the GMAT, right?
Right. Unless it is explicitly stated otherwise, different variables CAN represent the same number.
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